INVESTMENTS | BODIE, KANE, MARCUS Index Models The Capital Asset Pricing Model Copyright © 2014...

INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS

Index Models

The Capital Asset Pricing Model

• Advantages of a single-factor model• Risk decomposition• Systematic vs. firm-specific

• Single-index model and its estimation• Optimal risky portfolio in the index model• Index model vs. Markowitz procedure

Chapter Overview

• Advantages• Reduces the number of inputs for diversification• Easier for security analysts to specialize

• Model

• βi = response of an individual security’s return to the common factor, m; measure of systematic risk

• m = a common macroeconomic factor• ei = firm-specific surprises

A Single-Factor Market

iiii emrEr

• Regression equation:

• Expected return-beta relationship:

Single-Index Model

tetRtR iMiii

Miii RERE

• Variance = Systematic risk + Firm-specific risk:

• Covariance = Product of betas × Market index risk:

Single-Index Model

2( , )i j i j MCov r r

iMii e2222

2Cov ,i j i j Mr r

• Correlation = Product of correlations with the market index

Single-Index Model

Corr ,

Corr , Corr ,

i j M i M j Mi j

i j i M j M

i M j M

r r r r

• Variance of the equally-weighted portfolio of firm-specific components:

• When n gets large, σ2(ep) becomes negligible and firm specific risk is diversified away

Index Model and Diversification

Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk

Coefficient βp

Figure 8.2 Excess Returns on HP and S&P 500

Figure 8.3 Scatter Diagram of HP, the S&P 500, and HP’s SCL

tetRtR HPPSHPHPHP 500&

Table 8.1 Excel Output: Regression Statistics

for the SCL of Hewlett-Packard

• Correlation of HP with the S&P 500 is 0.7238

• The model explains about 52% of the variation in HP

• HP’s alpha is 0.86% per month (10.32% annually) but it is not statistically significant

• HP’s beta is 2.0348, but the 95% confidence interval is 1.43 to 2.53

Table 8.1 Interpreting the Output

Figure 8.4 Excess Returns on Portfolio Assets

• Alpha and Security Analysis1. Use macroeconomic analysis to estimate the risk

premium and risk of the market index.2. Use statistical analysis to estimate the beta

coefficients of all securities and their residual variances, σ2(ei).

3. Establish the expected return of each security absent any contribution from security analysis.

4. Use security analysis to develop private forecasts of the expected returns for each security.

Portfolio Construction and the Single-Index Model

• Single-Index Model Input List1. Risk premium on the S&P 500 portfolio2. Estimate of the SD of the S&P 500 portfolio3. n sets of estimates of• Beta coefficient• Stock residual variances• Alpha values

• Optimal risky portfolio in the single-index model• Expected return, SD, and Sharpe ratio:

• Optimal risky portfolio in the single-index model is a combination of• Active portfolio, denoted by A• Market-index portfolio, the passive

portfolio, denoted by M

• Optimal risky portfolio in the single-index model• Modification of active portfolio position:

* 01,A A Aw w

01 (1 )A

* 01,A A Aw w

• The Information Ratio• The Sharpe ratio of an optimally constructed risky

portfolio will exceed that of the index portfolio (the passive strategy):

• The Information Ratio• The contribution of the active portfolio depends

on the ratio of its alpha to its residual standard deviation

• The information ratio measures the extra return we can obtain from security analysis

Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance

Matrix

Table 8.2 Portfolios from the Single-Index and Full-Covariance Models

• Full Markowitz model may be better in principle, but• Using the full-covariance matrix invokes

estimation risk of thousands of terms• Cumulative errors may result in a portfolio that is

actually inferior to that derived from the single-index model

• The single-index model is practical and decentralizes macro and security analysis

Is the Index Model Inferior to the

Full-Covariance Model?

• Use 60 most recent months of price data • Use S&P 500 as proxy for M• Compute total returns that ignore dividends• Estimate index model without excess returns:

*ebrar m

Industry Version of the Index Model

• Adjust beta because• The average beta over all securities is 1; thus, the

best forecast of the beta would be that it is 1• Firms may become more “typical” as they age,

causing their betas to approach 1

Industry Version of the Index Model

Table 8.4 Industry Betas and Adjustment Factors

The Capital Asset Pricing Model

Capital Asset Pricing Model (CAPM)

• It is the equilibrium model that underlies all modern financial theory

• Derived using principles of diversification with simplified assumptions

• Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Assumptions

• Investors optimize portfolios a la Markowitz

• Investors use identical input list for efficient frontier

• Same risk-free rate, tangent CAL and risky portfolio

• Market portfolio is aggregation of all risky portfolios and has same weights

Resulting Equilibrium Conditions

• All investors will hold the same portfolio for risky assets – market portfolio

• Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Figure 9.1 The Efficient Frontier and the Capital Market Line

Market Risk Premium

•The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:

E(RM) = Ᾱσ2M

Where σ2M is the variance of the market

portfolio and Ᾱ is the average degree of risk aversion across investors

Return and Risk For Individual Securities

• The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio.

• An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio.

GE Example

• Covariance of GE return with the market portfolio:

• Therefore, the reward-to-risk ratio for investments in GE would be:𝐺𝐸′𝑠 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝐺𝐸′𝑠 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑤𝐺𝐸𝐸(𝑅𝐺𝐸)𝑤𝐺𝐸𝐶𝑜𝑣(𝑅𝐺𝐸,𝑅𝑀) = 𝐸(𝑅𝐺𝐸)𝐶𝑜𝑣(𝑅𝐺𝐸,𝑅𝑀)

𝑤𝑖𝐶𝑜𝑣 (𝑅𝑖𝑛

𝑖=1 ,𝑅𝐺𝐸) = 𝐶𝑜𝑣 (𝑤𝑖𝑅𝑖𝑛

𝑖=1 ,𝑅𝐺𝐸) = 𝐶𝑜𝑣൭ 𝑤𝑖𝑅𝑖𝑛

𝑖=1 ,𝑅𝐺𝐸൱

GE Example

• Reward-to-risk ratio for investment in market portfolio:

• Reward-to-risk ratios of GE and the market portfolio should be equal:

𝑀𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝐸(𝑅𝑀)𝜎𝑀2

𝐸(𝑅𝐺𝐸)𝐶𝑜𝑣(𝑅𝐺𝐸,𝑅𝑀) = 𝐸(𝑅𝑀)𝜎𝑀2

GE Example

• The risk premium for GE:

• Restating, we obtain:

fMGEfGE rrErrE

𝐸ሺ𝑅𝐺𝐸ሻ= 𝐶𝑜𝑣(𝑅𝐺𝐸,𝑅𝑀)𝜎𝑀2 𝐸(𝑅𝑀)

Expected Return-Beta Relationship

• CAPM holds for the overall portfolio because:

• This also holds for the market portfolio:

( ) ( ) andP k kk

E r w E r

( ) ( )M f M M fE r r E r r

Figure 9.2 The Security Market Line

Figure 9.3 The SML and a Positive-Alpha Stock

Single-Index Model and Realized Returns

• To move from expected to realized returns, use the index model in excess return form:

• The index model beta coefficient is the same as the beta of the CAPM expected return-beta relationship.

i i i M iR R e

Assumptions of the CAPM

• Individuals• Mean-variance optimizers• Homogeneous expectations• All assets are publicly traded

• Markets• All assets are publicly held • All information is available• No taxes• No transaction costs

Extensions of the CAPM

• Zero-Beta Model• Helps to explain positive alphas on

low beta stocks and negative alphas on high beta stocks

• Consideration of labor income and non-traded assets

Extensions of the CAPM

• Merton’s Multiperiod Model and hedge portfolios

• Incorporation of the effects of changes in the real rate of interest and inflation

• Consumption-based CAPM• Rubinstein, Lucas, and Breeden• Investors allocate wealth between

consumption today and investment for the future

Liquidity and the CAPM

• Liquidity: The ease and speed with which an asset can be sold at fair market value

• Illiquidity Premium: Discount from fair market value the seller must accept to obtain a quick sale. • Measured partly by bid-asked

spread• As trading costs are higher, the

illiquidity discount will be greater.

Figure 9.5 The Relationship Between Illiquidity and Average Returns

Liquidity Risk

• In a financial crisis, liquidity can unexpectedly dry up.

• When liquidity in one stock decreases, it tends to decrease in other stocks at the same time.

• Investors demand compensation for liquidity risk• Liquidity betas

CAPM and World

• Academic world• Cannot observe all tradable assets• Impossible to pin down market portfolio• Attempts to validate using regression analysis

• Investment Industry• Relies on the single-index CAPM model• Most investors don’t beat the index portfolio

INVESTMENTS | BODIE, KANE, MARCUS Index Models The Capital Asset Pricing Model Copyright © 2014...

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